Divergent Complex Sequence/Examples/(2 over 3 + 3i over 4)^n
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Example of Divergent Complex Sequence
Let $\sequence {z_n}$ be the complex sequence defined as:
- $z_n = \paren {\dfrac 2 3 + \dfrac {3 i} 4}^n$
Then $\ds \lim_{n \mathop \to \infty} z_n$ does not exist.
Proof
\(\ds \cmod {z_n}^2\) | \(=\) | \(\ds \cmod {\dfrac 2 3 + \dfrac {3 i} 4}^{2 n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac 4 9 + \dfrac 9 {16} }^n\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac {145} {144} }^n\) | ||||||||||||
\(\ds \) | \(\to\) | \(\ds \infty\) | as $\cmod {\dfrac 2 3 + \dfrac {3 i} 4} > 1$ |
Thus $\cmod {z_n} \to \infty$ and so the limit does not exist.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Example $\text {(ii)}$